The twin suicide attacks in the city of Volgograd are a calculated challenge to Vladimir Putin on the eve of the 2014 Sochi Winter Olympics. They are also a further depressing sign that the Kremlin's decade-long strategy for pacifying the North Caucasus has failed.

Can the Russian authorities guarantee security at the Games? The answer is no. Sunday and Monday's bombings – 32 dead and more than 100 injured so far – tell their own grim story.

These latest attacks could have been foreseen. Over the summer Russia's chief insurgent leader, Doku Umarov, gave a chilling warning. Umarov had declared an 18-month moratorium on targets in European Russia, which coincided with the rise of mass street protests against Vladimir Putin's rule. In a four-minute video clip released in July, however, Umarov announced a new, violent campaign against Russian "unbelievers". More specifically, he threatened to blow up Sochi.

Since 2007, Umarov and his followers have been fighting for an Islamic emirate across Russia's North Caucasus. For the jihadists, Sochi is a place of mournful ghosts: the Black Sea region was once home to the Muslim Circassians, who were driven out, murdered and deported to Turkey by the Russian army back in 1864. Some of the victims are buried in graves near the site of the key Olympic mountain-sports complex, in Krasnaya Polyana. In his video address, shot in a forest, Umarov accused Moscow of holding the Games "on the bones of many, many Muslims killed". The event was "satanic", he added.

Some analysts had doubted that Umarov's band of jihadist rebels had the capacity or numbers to carry out high-profile attacks. The two Volgograd bombings – one at the railway station, the other on a crowded commuter trolley bus – show the insurgents are indeed capable of striking outside their usual theatre of operations.

Russian media, meanwhile, reported that the bomber who blew up himself and the bus on Monday morning was an ethnic Russian and 32-year-old convert to Islam called Pavel Pechenkin. This is the Kremlin's worst nightmare: Slavic jihadists wreaking havoc.

According to Andrei Soldatov, an expert on Russia's security services, the bombings were designed to divert attention and police resources away from Sochi. The blasts could be a diversion before a further possible attack on the Olympics itself or Moscow, he said. Either way, the attacks have created a mood of panic. In the capital, the fearful atmosphere is reminiscent of March 2010, after two female suicide bombers blew themselves up on the Moscow metro, killing 40 people.

Cerwyn Moore, a senior lecturer at the University of Birmingham, who writes on terrorism and insurgency in the North Caucasus, said: "This certainly appears to be the work of Umarov's Caucasian Emirate. It's an interesting shift in tactics, moving towards soft targets outside the secure zone within the Olympic park itself."

This looked like the beginning of a sustained terror campaign, he said. He added: "I wouldn't be surprised if more attacks follow."

The authorities have taken extraordinary measures to safeguard the Sochi Games. Over the past four of five months, security forces have conducted numerous sweeps of mountain areas used by the rebels in Dagestan and other simmering republics. They have targeted known bomb-makers and facilitators. In December Ramzan Kadyrov, Chechnya's thuggish pro-Moscow leader, announced that Umarov was dead. The Russian military has even flown drones over suspected rebel hideouts.

The resort of Sochi, meanwhile, is under military lockdown, with all visitors checked, transport restricted, roadblocks in place, security sweeps carried out and political protests banned. The Federal Security Service – the main KGB successor agency – is conducting mass electronic surveillance. It will gather metadata from all Olympic participants, including spectators, judges, and athletes.

But will all this be enough? The Sochi Olympic zone may be impregnable. But if previous jihadist tactics are a guide, the rebels will seek out other soft targets.

More broadly, the attacks demonstrate the ineffectiveness of the Kremlin's counter-terrorism strategy. Over the past two decades, the insurgency in the North Caucasus has changed from a largely secular and nationalist uprising centred on Chechnya to an explicitly Islamist struggle encompassing all of Russia's southern Muslim republics. This war is characterised by daily gun battles between rebels and Russian security forces, local and federal. Mostly, it is confined to the North Caucasus itself. But as Volgograd, shows no part of Russia is entirely safe.

Brutal counter-insurgency operations have alienated local populations and fuelled militancy among the disaffected young. Poverty, joblessness, and corruption are further elements in a combustible mix.

Writing on Monday, the novelist and opposition figurehead Boris Akunin said the authorities needed to think about "horizontal" as well as "vertical" solutions to the deep-rooted problems of the North Caucasus. He called for dialogue and the "consolidation of society".

With just six weeks to go until the opening Olympic ceremony, time for fresh thinking is rapidly running out.